Ikeda map

In chaos theory, the Ikeda map is a discrete-time dynamical system that produces a strange attractor. It was introduced in 1979 by the physicist Kensuke Ikeda as a model for the behavior of light within a nonlinear optical resonator.^[1] The map demonstrates how a simple set of rules can lead to complex, chaotic behavior through a process of repeated rotation, scaling, and translation—a "stretch and fold" operation common in chaotic systems.

The map is defined by an iterative function on the complex plane. For a given complex number ${\displaystyle z_{n}}$, the next value is calculated as:$${\displaystyle z_{n+1}=A+Bz_{n}e^{i(|z_{n}|^{2}+C)}}$$Here, ${\displaystyle z_{n}}$ represents the electric field in the resonator at step ${\displaystyle n}$. The parameters ${\displaystyle A}$ and ${\displaystyle C}$ relate to the external laser light and the phase of the system, while ${\displaystyle B}$ (where ${\displaystyle B\leq 1}$) is a dissipation parameter representing energy loss in the resonator.^[2]

A commonly studied real-valued version of the map is given by the two-dimensional equations:$${\displaystyle x_{n+1}=1+u(x_{n}\cos t_{n}-y_{n}\sin t_{n}),\,}$$$${\displaystyle y_{n+1}=u(x_{n}\sin t_{n}+y_{n}\cos t_{n}),}$$where ${\displaystyle u}$ is a parameter and$${\displaystyle t_{n}=0.4-{\frac {6}{1+x_{n}^{2}+y_{n}^{2}}}.}$$For values of the parameter ${\displaystyle u\geq 0.6}$, this system exhibits chaotic behavior, generating the characteristic fractal attractor shown in the article's images.